The hashing bound for a Pauli channel defined by the probability mass function \(\mathbf {p} = (p_\mathrm {I},p_\mathrm {x},p_\mathrm {y},p_\mathrm {z})\)  is given by [1]}
\(C_\mathrm {H}(\mathbf {p}) = 1 - \mathrm {H}_2(\mathbf {p}).\) 
